1.Introduction HuilingWu Time-ConsistentStrategiesforaMultiperiodMean-VariancePortfolioSelectionProblem ResearchArticle

(1)

Volume 2013, Article ID 841627,13pages http://dx.doi.org/10.1155/2013/841627

Research Article

Time-Consistent Strategies for a Multiperiod Mean-Variance Portfolio Selection Problem

Huiling Wu

China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

Correspondence should be addressed to Huiling Wu; [email protected] Received 1 January 2013; Accepted 28 February 2013

Academic Editor: Francis T. K. Au

Copyright © 2013 Huiling Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It remained prevalent in the past years to obtain the precommitment strategies for Markowitz’s mean-variance portfolio optimization problems, but not much is known about their time-consistent strategies. This paper takes a step to investigate the time-consistent Nash equilibrium strategies for a multiperiod mean-variance portfolio selection problem. Under the assumption that the risk aversion is, respectively, a constant and a function of current wealth level, we obtain the explicit expressions for the time-consistent Nash equilibrium strategy and the equilibrium value function. Many interesting properties of the time-consistent results are identified through numerical sensitivity analysis and by comparing them with the classical pre-commitment solutions.

1. Introduction

Since the pioneering work of [1] in a single period, mean- variance formulation has been one of focused topics of portfolio selection optimization and has stimulated hundreds of extensions and applications. The interested readers can refer to [2,3] for detailed information. The objective of quite a number of existing mean-variance portfolio selection models is seeking an optimal strategy 𝜋(⋅) which maximizes the mean-variance utility𝐸[𝑋^𝜋_𝑇] − 𝜔Var[𝑋^𝜋_𝑇], where𝑋^𝜋_𝑇 is the terminal wealth. But it is well known that this mean-variance criterion lacks of iterated-expectation property, which gives rise to time-inconsistent investment strategy in the sense that Bellman optimality principle is not available any more.

The so-called time-inconsistent strategy means that optimal strategy obtained at time 𝑛 does not agree with optimal strategy derived at time 𝑚 where 𝑚 > 𝑛. Therefore, the optimal strategy in the classical time-inconsistent models is just optimal from the viewpoint of the initial time, and decision makers at any time 𝑘 after the initial time must commit themselves to the initial optimal strategy even if it is not optimal at time𝑘. So, the time-inconsistent optimal strategy in the classical mean-variance model is called the precommitment strategy. But this precommitment has been

criticized for lacking rationality. For one simple example, investment psychology and tastes will often change over time, and the decision maker at later time may not commit themselves to following a strategy which is not optimal at their current time. The work in [4] analyzed the incentives which induce the investor to revise her optimal strategy at subsequent dates under mean-variance criterion.

For this reason, we want to find an optimal strategy with time consistency which is necessary for a rational individual. The analysis of inconsistency can be traced back to [5] which pointed out that “optimal plan of the present moment is generally one which will not be obeyed” and the time-inconsistent problem can be solved by precommitment strategy or alternatively time-consistent strategy.

The authors of [5, 6] devoted themselves to identifying an intertemporal consumption programme that would be “the best plan that an agent would actually follow.” The work in [7] questioned the generality of the existence of Strotz- Pollak equilibrium and gave an alternative criterion of Nash equilibrium. More recently, it is of interest to study time- inconsistent problems. The work of [8, 9] investigated a time-consistent strategy for a consumption and investment problem with nonexponential discounting. The work in [10] gave general approaches to handle time-inconsistent

(2)

problems by viewing them as a game theoretic framework and looking for Nash subgame perfect equilibrium points. They formally defined the continuous time equilibrium concept and derived the extension of HJB equation and its verification theorem for a very general objective functions. The work in [11] studied a continuous-time mean-variance portfolio optimization model on the assumption that the risk aversion factor depended dynamically on the current wealth. In view of the extension of HJB equation developed in [10], they obtained the time-consistent equilibrium control and equilibrium value function. The work in [12] provided explicit solutions to a series of cases, including mean-standard deviation in continuous-time setting. The work in [13] investigated optimal mean-variance time-consistent investment and reinsurance policies for an insurer under continuous- time setting. The work in [14] developed a fully numerical scheme to determine time-consistent mean-variance strategy based on piecewise constant policy technique. As for the discrete-time mean-variance models, the work in [15] gave a complicated backwards recursive relationship about time- consistent investment strategy but had not found analytical expression for the strategy.

To the best of our knowledge, no existing literature has given time-consistent equilibrium strategy and equilibrium value function in closed form for discrete-time mean- variance asset allocation. Our research will fill the gap. We view this decision-making process as a noncooperative game and suppose that there is one decision maker, referred to as

“decision maker𝑛”, for each point of time𝑛. This assumption is reasonable. On one hand, in the real world, there are often quite different persons who will join in the decision-making process especially when the investment horizon is long; on the other hand, we can image decision-maker𝑛as the future incarnation of themselves at time𝑛considering that the tastes of the decision maker will change over time. So our work in this paper is listed as follows: (a) derive the analytical expressions for the time-consistent equilibrium strategy and equilibrium value function when the risk aversion is assumed to be a constant and a function of current wealth, respectively;

(b) when risk aversion factor is a constant, compare our time-consistent results with the precommitment ones in [2]

and present the particular properties of the time-consistent results;(c)study the problem in discrete-time setting with nonconstant risk aversion which is a function of current wealth and identify the properties of the investment proportion by numerical analysis.

The rest of the paper is organized as follows. InSection 2, the problem formulation is presented, and the recursive formula of the equilibrium value function is derived. In Section 3, equilibrium strategy and equilibrium value function for mean-variance model with constant risk aversion are obtained. Comparison of our time-consistent results with the precommitment ones in [2] is also given in this section.

InSection 4, the equilibrium results are investigated on the assumption that the risk aversion depends dynamically on the current wealth and numerical analysis is given to demonstrate the properties of investment proportion.Section 5presents our conclusions.

2. Problem Formulation

In this paper, we assume that investors join the market at time 0with an initial wealth𝑥₀and plan to process the investment in𝑇consecutive time periods. There are one risk-free asset and𝑚₁risky assets in the market. The𝑚₁risky assets have random returns𝑅_𝑛= (𝑅_𝑛,1, 𝑅_𝑛,2, . . . , 𝑅_𝑛,𝑚₁)^󸀠at period𝑛(time interval [𝑛, 𝑛 + 1)) where 𝑅_𝑛,𝑘 denotes the random return of the 𝑘th asset at period 𝑛and superscript “^󸀠” stands for the transpose of a matrix or vector. Denote by𝑟_𝑛^𝑓the return of the risk-free asset at period 𝑛, and denote by 𝑋_𝑛 and 𝜋_𝑛 = (𝜋_𝑛,1, . . . , 𝜋_𝑛,𝑚₁)^󸀠the wealth available for investment and the amounts invested in𝑚₁risky asset at time𝑛, respectively.

Then the wealth dynamics is

𝑋_𝑛+1= (𝑋_𝑛−∑^𝑚¹

𝑘=1

𝜋_𝑛,𝑘) 𝑟^𝑓_𝑛 +∑^𝑚¹

𝑘=1

𝜋_𝑛,𝑘𝑅_𝑛,𝑘= 𝑋_𝑛𝑟_𝑛^𝑓+ 𝜋^󸀠_𝑛𝑅^𝑒_𝑛, (1)

where𝑅^𝑒_𝑛,𝑘 = 𝑅_𝑛,𝑘− 𝑟_𝑛^𝑓, 𝑅_𝑛^𝑒 = (𝑅^𝑒_𝑛,1, 𝑅^𝑒_𝑛,2, . . . , 𝑅^𝑒_𝑛,𝑚₁)^󸀠for𝑛 = 0, 1, . . . , 𝑇 − 1.

As we know, the classic mean-variance optimization problem is as follows:

{^𝜋0,𝜋max1,...,𝜋𝑇−1}𝐸 (𝑋^𝜋_𝑇) − 𝜔Var(𝑋^𝜋_𝑇) , (2)

which results in a time-inconsistent strategy, that is, precommitment strategy. Therefore, as mentioned inSection 1, this paper aims to solve this problem from another perspective and to look for the time-consistent Nash equilibrium investment strategy. To this end, we first give the definition of Nash equilibrium strategy according to [10] and the references therein.

Let𝜋(𝑛) = {𝜋_𝑛, 𝜋_𝑛+1, . . . , 𝜋_𝑇−1}be the policy made at time 𝑛and

𝐽_𝑛(F_𝑛, 𝜋 (𝑛)) = 𝐸 (𝑋^𝜋(𝑛)_𝑇 |F_𝑛) − 𝜔 (F_𝑛)Var(𝑋^𝜋(𝑛)_𝑇 |F_𝑛) , (3)

where 𝑋^𝜋(𝑛)_𝑇 is the terminal wealth corresponding to the investment strategy𝜋, andF_𝑛 is the information at time𝑛, such as wealth level. A natural assumption is that risk aversion 𝜔is a function ofF_𝑛.

Definition 1. Let ̂𝜋 be a fixed control law. For an arbitrary point𝑛 (𝑛 = 0, 1, . . . , 𝑇 − 1), one selects an arbitrary control value𝜋_𝑛and define the strategy𝜋(𝑛) = (𝜋_𝑛, ̂𝜋_𝑛+1, . . . , ̂𝜋_𝑇−1).

Then̂𝜋is said to be a subgame perfect Nash equilibrium strategy (or simply equilibrium strategy) if for all𝑛 < 𝑇, it satisfies

max𝜋𝑛 𝐽_𝑛(F_𝑛; 𝜋 (𝑛)) = 𝐽_𝑛(F_𝑛; ̂𝜋 (𝑛)) . (4)

(3)

In addition, if equilibrium strategŷ𝜋exists, the equilibrium value function is defined as

𝑉_𝑛(F_𝑛) = 𝐽_𝑛(F_𝑛; ̂𝜋 (𝑛)) . (5)

Let̂𝜋(𝑛)be the Nash equilibrium strategy at time𝑛, then Definition 1 makes it possible to solve the problem by the following procedure:

(a) ̂𝜋(𝑇 − 1) = ̂𝜋_𝑇−1 = arg max_𝜋_𝑇−1[𝐸(𝑋^𝜋_𝑇^𝑇−1 | F_𝑇−1) − 𝜔(F_𝑇−1)Var(𝑋^𝜋_𝑇^𝑇−1 |F_𝑇−1)];

(b) given that the decision maker𝑇 − 1 will use ̂𝜋_𝑇−1,

̂𝜋_𝑇−2is the optimal control that optimizes objective function𝐽_𝑇−2(F_𝑇−2; (𝜋_𝑇−2, ̂𝜋_𝑇−1));

(c) generally,̂𝜋_𝑛 is obtained by letting decision maker𝑛 choose𝜋_𝑛to maximize𝐽_𝑛given that the forthcoming decision makers𝑛+1, . . . , 𝑇−1will choose the strategy

̂𝜋(𝑛 + 1) = (̂𝜋_𝑛+1, . . . , ̂𝜋_𝑇−1); that is,

̂𝜋_𝑛=arg max

𝜋𝑛 𝐽_𝑛(F_𝑛; (𝜋_𝑛, ̂𝜋_𝑛+1, . . . , ̂𝜋_𝑇−1)) . (6) Now we try to derive the time-consistent Nash equilibrium strategy and value function, but first we need to give the following notations and assumptions throughout this paper:

(N1)𝑟_𝑛^𝑒 = 𝐸(𝑅^𝑒_𝑛), 𝑛 = 0, 1, . . . , 𝑇 − 1;

(N2)𝜎_𝑛 = 𝐸(𝑅^𝑒_𝑛𝑅^𝑒_𝑛^󸀠) − 𝑟^𝑒_𝑛𝑟_𝑛^𝑒^󸀠, 𝑛 = 0, 1, . . . , 𝑇 − 1;

(N3)𝜅_𝑛= 𝑟^𝑒_𝑛^󸀠(𝜎_𝑛)⁻¹𝑟_𝑛^𝑒, 𝑛 = 0, 1, . . . , 𝑇 − 1;

(N4)𝑔^𝜋(𝑛)_𝑛 (F_𝑛) = 𝐸[𝑋^𝜋(𝑛)_𝑇 | F_𝑛], 𝑔_𝑛(F_𝑛) = 𝐸[𝑋_𝑇^̂𝜋(𝑛) | F_𝑛], 𝑛 = 0, 1, . . . , 𝑇 − 1;

(N5)ℎ^𝜋(𝑛)_𝑛 (F_𝑛) = 𝐸[(𝑋^𝜋(𝑛)_𝑇 )² | F_𝑛], ℎ_𝑛(F_𝑛) = 𝐸[(𝑋^̂𝜋(𝑛)_𝑇 )² | F_𝑛], 𝑛 = 0, 1, . . . , 𝑇 − 1.

(A1) The distribution function of the random returns𝑅_𝑛 is𝐹_𝑛, and {𝑅_𝑛, 𝑛 = 0, 1, . . . , 𝑇 − 1}is assumed to be statistically independent.

(A2)𝜎_𝑛is assumed to be positive definite.

(A3) Short selling is allowed for all risky assets in all periods. Unlimited borrowing and lending are permitted.

Transaction costs are not taken into account.

(A4) Capital additions or withdrawals are forbidden for all assets in all periods.

With the notations above, we can obtain the recursions of 𝐽_𝑛and𝑉_𝑛. For the sake of convenience, we define𝐸_𝑛,F_𝑛(⋅) = 𝐸[⋅ |F_𝑛]and Var_𝑛,F_𝑛(⋅) =Var(⋅ |F_𝑛), then

𝐽_𝑛(F_𝑛; 𝜋 (𝑛))